Subgroups — C₂×C₁₃×#2255 #15720066

Order 3328.

518 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#499	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2255	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#367	C₂	1	2	yes	maximal
5	C₂×C₁₃×#366	C₂	1	2	yes	maximal
6	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
8	C₂×C₁₃×#125	—	2	1	no	—
9	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
10	C₁₃×#500	—	2	2	no	—
11	C₁₃×#500	C₂²	1	4	yes	—
12	C₁₃×#499	—	2	2	no	—
13	C₁₃×#499	C₂²	1	4	yes	—
14	C₂×C₁₃×#107	—	2	1	no	—
15	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#107	C₂²	1	2	yes	—
18	C₂×C₁₃×#106	—	2	1	no	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#367	C₂²	1	8	yes	—
21	C₁₃×#366	C₂²	1	8	yes	—
22	C₁₃×#346	C₂²	1	4	yes	—
23	C₂²×Q₈×C₁₃	—	4	1	no	—
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₁₃×C₄⋊Q₈	—	2	4	no	—
27	C₁₃×C₄⋊Q₈	—	4	2	no	—
28	C₁₃×#125	—	2	4	no	—
29	C₁₃×#125	—	4	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
34	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
35	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₁₃×C₁₆	Q₈	1	8	yes	complemented
38	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
39	C₁₃×#107	—	2	4	no	—
40	C₁₃×#107	—	2	1	no	—
41	C₁₃×#107	—	2	1	no	—
42	C₁₃×#107	—	2	2	no	—
43	C₁₃×#107	C₂³	1	2	yes	—
44	C₁₃×#107	C₂³	1	4	yes	—
45	C₁₃×#107	C₂³	1	2	yes	—
46	C₁₃×#106	—	2	4	no	—
47	C₄×C₈×C₁₃	D₄	1	2	yes	—
48	C₄×C₈×C₁₃	C₂³	1	2	yes	—
49	C₂×#2255	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×Q₈×C₁₃	—	4	4	no	—
51	C₂×Q₈×C₁₃	—	8	4	no	—
52	C₂²×C₄×C₁₃	—	4	1	no	—
53	C₂²×C₄×C₁₃	—	4	1	no	—
54	C₂²×C₄×C₁₃	—	4	2	no	—
55	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	Q₁₆	1	4	yes	—
59	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
62	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	—	2	2	no	—
64	C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₁₃×C₄⋊C₄	—	2	4	no	—
66	C₁₃×C₄⋊C₄	—	4	1	no	—
67	C₁₃×C₄⋊C₄	—	4	1	no	—
68	C₁₃×C₄⋊C₄	—	4	2	no	—
69	C₁₃×C₄⋊C₄	—	4	4	no	—
70	C₁₃×C₄⋊C₄	—	4	4	no	—
71	C₁₃×C₄⋊C₄	—	4	4	no	—
72	C₄²×C₁₃	D₈	1	2	yes	—
73	C₄²×C₁₃	C₂×D₄	1	2	yes	—
74	C₁₃×C₁₆	C₂×Q₈	1	8	yes	complemented
75	C₂×#500	C₂×C₁₃	1	1	yes	characteristic
76	C₂×#499	C₂×C₁₃	1	1	yes	characteristic
77	#2255	C₂×C₁₃	1	8	yes	direct product factor
78	C₂×#367	C₂×C₁₃	1	2	yes	—
79	C₂×#366	C₂×C₁₃	1	2	yes	—
80	C₂×#346	C₂×C₁₃	1	1	yes	characteristic
81	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
82	Q₈×C₁₃	—	8	8	no	—
83	C₂×C₄×C₁₃	—	2	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	4	2	no	—
86	C₂×C₄×C₁₃	—	4	2	no	—
87	C₂×C₄×C₁₃	—	4	4	no	—
88	C₂×C₄×C₁₃	—	8	2	no	—
89	C₂×C₄×C₁₃	—	8	4	no	—
90	C₂×C₄×C₁₃	—	8	2	no	—
91	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
96	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
97	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
98	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
99	C₈×C₁₃	C₂²×Q₈	1	2	yes	—
100	C₂×C₄⋊Q₈	—	2	1	no	—
101	C₂×#125	—	2	1	no	—
102	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
103	#500	—	2	2	no	—
104	#500	C₂²×C₁₃	1	4	yes	—
105	#499	—	2	2	no	—
106	#499	C₂²×C₁₃	1	4	yes	—
107	C₂×#107	—	2	1	no	—
108	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
109	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
110	C₂×#107	C₂²×C₁₃	1	2	yes	—
111	C₂×#106	—	2	1	no	—
112	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
113	#367	C₂²×C₁₃	1	8	yes	—
114	#366	C₂²×C₁₃	1	8	yes	—
115	#346	C₂²×C₁₃	1	4	yes	—
116	C₂²×C₁₃	#500	1	2	yes	—
117	C₂²×C₁₃	#509	1	2	yes	—
118	C₂²×C₁₃	#510	1	2	yes	—
119	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	—	8	2	no	—
123	C₄×C₁₃	—	8	4	no	—
124	C₄×C₁₃	—	8	2	no	—
125	C₄×C₁₃	#500	1	2	yes	—
126	C₄×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
127	C₄×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
128	C₂²×Q₈	—	4	1	no	—
129	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
131	C₄⋊Q₈	—	2	4	no	—
132	C₄⋊Q₈	—	4	2	no	—
133	#125	—	2	4	no	—
134	#125	—	4	2	no	—
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄⋊C₄	—	2	1	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	4	1	no	—
139	C₂×C₄⋊C₄	—	4	1	no	—
140	C₂×C₄⋊C₄	—	4	1	no	—
141	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
142	C₂×C₁₆	Q₈×C₁₃	1	8	yes	complemented
143	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
144	#107	—	2	4	no	—
145	#107	—	2	1	no	—
146	#107	—	2	1	no	—
147	#107	—	2	2	no	—
148	#107	C₂³×C₁₃	1	2	yes	—
149	#107	C₂³×C₁₃	1	4	yes	—
150	#107	C₂³×C₁₃	1	2	yes	—
151	#106	—	2	4	no	—
152	C₄×C₈	D₄×C₁₃	1	2	yes	—
153	C₄×C₈	C₂³×C₁₃	1	2	yes	—
154	C₂×C₁₃	#2255	1	4	yes	direct product factor
155	C₂×C₁₃	C₂×#500	1	1	yes	characteristic
156	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
157	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
158	C₂×Q₈	—	4	4	no	—
159	C₂×Q₈	—	8	4	no	—
160	C₂²×C₄	—	4	1	no	—
161	C₂²×C₄	—	4	1	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
164	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
166	C₂×C₈	C₁₃×Q₁₆	1	4	yes	—
167	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
168	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
169	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
170	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	4	no	—
174	C₄⋊C₄	—	4	1	no	—
175	C₄⋊C₄	—	4	1	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	4	4	no	—
179	C₄⋊C₄	—	4	4	no	—
180	C₄²	C₁₃×D₈	1	2	yes	—
181	C₄²	C₂×D₄×C₁₃	1	2	yes	—
182	C₁₆	C₂×Q₈×C₁₃	1	8	yes	complemented
183	C₁₃	C₂×#2255	1	1	yes	characteristic, direct product factor, Sylow
184	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
185	Q₈	—	8	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	—	8	2	no	—
194	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
195	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
196	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
197	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
198	C₂×C₄	C₂×C₁₃×Q₁₆	1	2	yes	—
199	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
200	C₈	C₁₃×C₄⋊Q₈	1	2	yes	—
201	C₈	C₂×C₁₃×Q₁₆	1	4	yes	—
202	C₈	C₂²×Q₈×C₁₃	1	2	yes	—
203	C₂²	C₁₃×#500	1	2	yes	—
204	C₂²	C₁₃×#509	1	2	yes	—
205	C₂²	C₁₃×#510	1	2	yes	—
206	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	8	2	no	—
210	C₄	—	8	4	no	—
211	C₄	—	8	2	no	—
212	C₄	C₁₃×#500	1	2	yes	—
213	C₄	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
214	C₄	C₂²×C₁₃×Q₁₆	1	1	yes	characteristic
215	C₂	C₁₃×#2255	1	4	yes	direct product factor
216	C₂	C₂×C₁₃×#500	1	1	yes	characteristic
217	C₂	C₂×C₁₃×#509	1	1	yes	characteristic
218	C₂	C₂×C₁₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₁₃×#2255	C₂	direct (N × H)
37. C₂×C₁₃×C₁₆	Q₈	semidirect
49. C₂×#2255	C₁₃	direct (N × H)
74. C₁₃×C₁₆	C₂×Q₈	semidirect
77. #2255	C₂×C₁₃	direct (N × H)
142. C₂×C₁₆	Q₈×C₁₃	semidirect
154. C₂×C₁₃	#2255	direct (N × H)
182. C₁₆	C₂×Q₈×C₁₃	semidirect
183. C₁₃	C₂×#2255	direct (N × H)
215. C₂	C₁₃×#2255	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×#500	2
C₂×C₁₃×#499	2
C₁₃×#2255	2
C₂×C₁₃×#367	2
C₂×C₁₃×#366	2
C₂×C₁₃×#346	2
C₂×#2255	13

Subgroups — C2×C13×#2255 #15720066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2255 #15720066